Circulatory Systems and Mortality Rates

bioRxiv preprint doi: https://doi.org/10.1101/2021.05.27.446029; this version posted May 28, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission.

Circulatory systems and mortality rates

Gurdip Uppal, Pinar Zorlutuna, Dervis Can Vural∗ Department of Physics, University of Notre Dame, USA ( Dated: May 27, 2021) Aging is a complex process involving multiple factors and subcellular processes, ultimately leading to the death of an organism. The microscopic processes that cause aging are relatively well understood and effective macroscopic theories help explain the universality of aging in complex systems. However, these theories fail to explain the diversity of aging observed for various lifeforms. As such, more complete “mesoscopic” theories of aging are needed, combining the biophysical details of microscopic failure and the macroscopic structure of complex systems. Here we explore two models: (1) a network theoretic model, and (2) a convection diffusion model emphasizing the biophysical details of communicated signals. The first model allows us to explore the effects of connectivity structures on aging. In our second model, cells interact through cooperative and antagonistic factors. We find by varying the ratio at which these factors affect cell death, as well as the reaction kinetics, diffusive and flow parameters, we obtain a wide diversity of mortality curves. As such, the connectivity structures as well as the biophysical details of how various factors are transported in an organism may explain the diversity of aging observed across different lifeforms.

INTRODUCTION organisms at the cellular level, it is unclear why they manifest differently on the macroscopic scale depending on the bau- In many complex organisms, including humans, the prob- plan. ability of death (mortality rate) µ(t) typically increases ex- Between these two extremes of evolutionary and biochem- ponentially with age t. This empirical trend, µ(t) ∼ eλt is ical theories, there are “mesoscopic” theories of aging, which known as the Gompertz law [1]. However, a broader look attempt to bridge cellular-level failures with organismic fail- across taxa reveals mortality curves that are as diverse as life ure. Reliability theory [13] views the organism as a collec- itself [2]. Most interestingly, there appears to be phyloge- tion of clusters (e.g. organs) of smaller units (e.g. cells), and netic correlations in the shape of µ(t) [3]: Mammals tend assumes that if all units within any one of the cluster fails, to have steep mortality curves that increase many folds dur- then so does the organism. The network theory of aging [14], ing their lifespan, which means that an old mammal is many views the organism as a large collection of units that depend times more probable to die compared to a mammal of aver- on one another to function. Once a unit malfunctions, so will age age. In contrast, for amphibians and reptiles µ(t) changes its dependents. As such, microscopic failures that otherwise little within their lifetime. Plants tend to age even less, and accumulate slowly, can suddenly cascade into a macroscopic some can even anti-age, i.e. exhibit mortality rates that de- catastrophe. crease with age. Invertebrates on the other hand, are scattered The goal of this paper is to expand the interdependence net- across the map: Some, such as water fleas and bdelloid ro- work theory of aging [14] to develop a new hypothesis that tifers age as rapidly as mammals, while others, such as the can account for the taxa-specific characteristics of mortality hydra and hermit crab do not seem to age. What is the reason curves. The main idea is that in a biophysically realistic in- behind these phylogenetic trends? What taxa-specific anatom- terdependence network, the structure of connections between ical features set the characteristics of the mortality curve of an the nodes are determined by the exchange of various types organism? of goods and signals, as mediated by diffusion and circula- Evolutionary theories explain aging as due to selection fail- tory flow. Therefore, the transport system within an organism ing to remove deleterious mutations whose effects manifest (by virtue of being a proxy for the interdependency between after peak fertility (mutation accumulation theory) [4, 5] or functional body components) will govern how it fails [15–17]. due to mutations that increase fitness and reproductive poten- We will argue that the qualitative differences in the mortality tial early in life at the cost of that later in life (antagonistic curves may be due to (a) the structure of the interdependence pleiotropy theory) [6–8]. These standard theories predict that network and (b) the type of goods exchanged through the net- mortality rate should rapidly increase after maturation, and work. therefore cannot account for the variations in mortality curves, The simplest species rely primarily on diffusion to trans- nor the mechanism behind phylogenetic correlations. port necessary resources into and waste out of the organ- On the other end of the spectrum there are mechanistic the- ism. As specialized organs form, complex species evolved ories of aging, seeking to understand aging in terms of bio- circulatory systems to more effectively transport required re- chemical basic principles such as telomere shortening [9], sources between components in the organism. This allows for cell damage and repair, [10], oxidative stress [11], and cell stronger interdependencies to form within an organism, lead- metabolism [12]. While these factors surely drive aging in all ing to stronger aging effects. For example, in a sparse decentralized interdependence network such as that of a bush or ivy, the function of a leaf or terminal branch in one location does not critically depend on another one far away. Because ∗ [email protected] of this decentralized structure, accumulated failures will only bioRxiv preprint doi: https://doi.org/10.1101/2021.05.27.446029; this version posted May 28, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 2

influence near-by nodes and not lead to a catastrophic cascade. (network nodes) shape the mortality curves of a species. The The bauplan of a mammal on the other hand is a dense, many- nodes in a network can represent genes, cells, organs or some to-many interdependence network. Virtually every functional collection of components that has a specialized function in an component couples to every other via blood stream, and there- organism; whereas edges between nodes represent dependen- fore the failure of few components will rapidly cascade into a cies. Each node is assumed to be in either a functional or catastrophe. Thus, our first specific hypothesis is that organ- nonfunctional state. The model is implemented as follows: isms with a large number of components that are functionally (1) we create a random directed network. We explore vari- coupled irrespective of the distance between them, will exhibit ous topologies as shown in Fig. 1. The scale free random steeper mortality curves. networks in Fig. 1d,e, were generated using the Barabasi- In the first part of our paper we will simulate the failure Albert model with a given initial number of connected nodes modes of cohorts of networks that vary in structure, to illus- m0. The random networks in Fig. 1e,f, were generated us- trate that dense, many-to-many networks (as in vertebrates) ing the Erdos-Renyi model varying the probability of an edge exhibit steep mortality curves, whereas sparse, few-to-few p. The fully connected network in Fig. 1h was generated by networks (as in plants) exhibit flatter or even decreasing mor- attaching one edge between any two nodes, with the direc- tality curves. tion chosen randomly. We also assign an initial fraction of Secondly we will argue that the type of products exchanged nodes, η to be in a nonfunctional state. (2) We age the net- between nodes make an important difference in mortality work system by first randomly turning functional nodes into curves. For example, in reptiles and amphibians, oxygenated non-functional ones with probability πd. We then go through and venous blood mixes, whereas in mammals, it does not. “repair” stage, where nonfunctional nodes turn into functional As such, for a mammal, the death of every cell is a small step nodes with probability πr. Next, we cascade the failures: all closer to organismic failure, whereas for a reptile or amphib- nodes whose majority of inputs are non-functional turns into a ian, this infinitesimal loss of function may be partly compen- non-functional node. We iterate this procedure of damaging, sated by lesser CO2 stress. repairing and cascading till only a small fraction of nodes τ Accordingly, in the second part of our paper we will exam- remain functional, after which the organism is considered to ine the mortality statistics of cell populations that exchange be dead. Throughout, we set τ = 0.01, corresponding to 1% only positive factors versus a mixture of positive and nega- of initial number of nodes. We also set πd = 0.002, and vary tive factors. We will do so by simulating the demographics η and πr. We study an ensemble of 10000 networks for each of a population of simplified organisms where cells are ar- topology and record the death times to determine the mortality ranged around a circulating fluid. In one population, clean rate over time (Fig. 1). and dirty blood will not mix, so cells will only be subject to Diffusion-advection model. In the second model, we study each other’s positive effects (growth factors, survival factors, the role of the type and transport-mode of signals being com- antioxidant enzymes [18–21]). We will see that this popula- municated, focusing on a simple schematic circular topology. tion will exhibit a rapid aging curve, i.e. mortality rate will be In this model, dependencies are mediated by diffusive and cir- many times more larger at average lifespan compared to that culatory processes. Cells interact through secreted coopera- at half of lifespan. This population will be compared with a tive factors and/or antagonistic factors. Positive factors help “mixed blood” one, where cells will be subject to each other’s other cells survive or function, whereas antagonistic factors positive as well as negative factors. We will see that this pop- harm or hinder cellular function. These factors then diffuse, ulation will exhibit weak aging characteristics, i.e. will have flow, and decay within the circulating fluid. We quantify these flatter mortality curves. model assumptions with the following system of equations, We find that the biophysical details such as type of factor, φk ψm reaction kinetics, and diffusion and flow rates, can drastically n˙ = −α γ 0 + (1 − γ) n (1) k k m m affect the aging dynamics of a system, even for a schematic φ0 + φ ψ + ψ0 ˙ 2 circular circulatory system. Our results emphasize the impor- φ = Dφ∇ φ − v · ∇φ − λφφ + Aφn (2) tance of biophysically accurate models in understanding the ˙ 2 interdependence structure of an organism and the aging dy- ψ = Dψ∇ ψ − v · ∇ψ − λψψ + Aψn (3) namics of complex living systems. Here n = n(x, t) denotes the cell concentration at time t and position x, φ(x, t) is the density of cooperative factors, and ψ(x, t) is the density for negative factors. Here it is assumed METHODS that the cell repair and death are influenced by φ and ψ according to the Michaelis-Mentin (Hill) functions. We hypothesize To explore the effects of the interdependency structure on the ratio of positive versus negative factors mixed in the cir- aging dynamics, we study two models of aging. One where culatory fluid will in general vary across species and have a the interdependency between body components are governed significant effect on the aging dynamics of that species. To by a complex network structure, and one where it is described study the effects of varying the ratio of positive over negative by a simple circulatory flow. factors, we introduce a mixing factor γ ∈ [0, 1] in equation Network model. In the first model, we simulate aging of 1. The constant α > 0 gives an overall scaling of the death interdependency networks that have a variety of structures to rate of cells. Note that the two terms on the right hand side of study how local and global couplings between components equation 1 could have had two independent coefficients a, b bioRxiv preprint doi: https://doi.org/10.1101/2021.05.27.446029; this version posted May 28, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 3

FIG. 1. Aging in various network topologies. a, Network where one central node is connected to all others. Except for the central node all other nodes are essentially independent. Individual nodes primarily die independently, unless the central node fails, which leads to a system-wide failure. Hence, the mortality curve is mostly flat. b, Nodes connected in a chain. With this topology, the failure of a node will cause all nodes “downstream” to fail as well. c, Network with branching topology where outer nodes depend on inner ones. The entire network depends on a central node either directly or indirectly. d, Scale free random network with low connectivity (m0 = 2). e, Scale free random network with high connectivity (m0 = 50). f, Random network with low connectivity (probability of a link, p = 0.1). g, Random network with high connectivity (probability of a link, p = 0.5) h, Fully connected randomly directed network. In general, networks that are less connected have flatter mortality curves, whereas more connected topologies lead to steeper aging curves. The fully connected topology (h) gives the steepest aging curves, and the “one-to-all” topology (a) gives the flattest curves. In addition to varying topologies, we explored varying initial conditions, by varying the intial fraction of nonfunctional nodes η, and varying repair rates πr. We see for more locally connected networks, the repair rate has a much larger effect. In panels c,d, we see that the repair rate determines the aging curve and the initial condition has little effect. Whereas for more globally connected networks e-h, a larger repair rate does little to shape the mortality curve. The initial condition on the other hand, has a larger effect in these cases and less so on locally connected networks. For the simple topologies of a,b, the mortality curves are predominately flat and nearly independent of parameters. Images of the different networks are shown with limited edges for clarity. For simulations, the branching network was chosen with 1023 nodes, and all other models were chosen with 1000 nodes. An ensemble of 10000 “organisms” was simulated for each case to determine mortality curves. Mortality curves were calculated using 1/4 the standard deviation of death times as a time step (δt) and taking a discrete derivative of the survival function (µ(t) ≈ −[S(t + δt) − S(t)]/[S(t)δt]) and then normalized by mean mortality rate over time. *The initial fraction of nonfunctional nodes is chosen to be η = 0.1 for the red and purple curves in chain topology (b), since much larger η leads to nearly immediate failure of the whole network. bioRxiv preprint doi: https://doi.org/10.1101/2021.05.27.446029; this version posted May 28, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 4

a instead of γ and 1 − γ, since setting α = a + b and γ = a+b difference from varying the repair rates πr (Fig. 1e-h). How- k m φ0 ψ ever, for sparser, locally connected networks (Fig. 1c,d), we leads to a k k + b ψm+ψm . φ0 +φ 0 see higher repair rates lead to flatter mortality curves. This The constants φ0 and ψ0 (which we set to 1 throughout) is because failure propagates slower in less connected topolo- give the saturation constants for positive φ(x, t) and negative gies, and is more amenable to repair. ψ(x, t) factors respectively. These correspond to the “required We therefore find that the connectivity structure, initial amount” of factors for a cell to function normally for positive conditions, and regenerative capabilities of an organism can factors, and a “tolerance threshold” for negative ones, above greatly influence its aging dynamics. Simpler organisms that which cells begin to die faster. The constants kφ and kψ de- are only locally connected age slower or do not age, whereas scribe the steepness of the response of cells to the correspond- more complex species with more global connectivity struc- ing factor. The constants D give the diffusion rate for each ture have much steeper mortality curves (Fig. 1). What can chemical (denoted by subscripts), v gives the flow rate, taken we say about differences in mortality patterns between species to be constant and the same for all factors, λ’s correspond to of similar complexity? From the network-theoretic viewpoint, decay rates, and A’s give the rate at which cells produce the reptiles and amphibians are also complex multicellular organ- factors. isms, yet we observe them to age much slower than many In our numerical simulations, we treat cells as discrete mammals. How do we then explain the differences in aging points that die stochastically with probability given by the across complex organisms, and what are the roles of diffusive term in square brackets in equation 1. The chemical fac- and advective processes? Below, we will see that in additional tors are treated as continuous fields that evolve according to to the topology of interactions, the type of interactions and the a finite difference scheme on a one-dimensional circle with mode of transport mediating the interactions also play an im- length L = 100, following periodic boundary conditions. portant role. More specifically, at each time step ∆t, cells secrete each factor at a rate given by Aφ for positive factors and Aψ for negative ones. Cells then die with probability given by h k m i Effects of hill constants and mixing on aging dynamics φ0 ψ p = α γ k k + (1 − γ) m m ∆t. The chemicals dif- φ0 +φ ψ +ψ0 fuse, flow, and decay according to a forward Euler finite dif- To study the effects of the types of signals communicated ference scheme using a central difference for diffusion and a in an organism, as well as their transport properties and re- first-order upwind scheme for advection. Lifespan is defined action kinetics, we next explore aging patterns of a group of as the time at which all cells are dead. Each system is run mul- cells situated around a circulating flow, which mediates their tiple times, recording the death time for each to give survival interactions. S(t) and mortality curves µ(t). We start by investigating the effects of the hill parameters and mixing factor γ on the aging dynamics of a species. We plot the cell population, total number of living organisms, and RESULTS mortality rate versus time, normalized by mean lifetime in Fig. 2. We vary both the hill constants and the mixing factor γ. Effects of topology and connectivity on aging For simplicity, we take the two hill parameters to be equal, kφ = kψ = k. We also set Dφ = Dψ = 5, α = 100, and set We first study the effects of different connectivity struc- flow rate to zero. Note that changing the constant α amounts tures on the aging of organisms. We explore the effects of to a global rescaling of time. Since we rescale time by the different initial conditions and bauplans by varying network mean lifetime, the constant α does not play a major role in topologies and the initial fraction of nonfunctional nodes η our results. (Fig. 1). For sparser, locally connected networks, we see the For each parameter set, we run 1000 simulations and record mortality curves are much flatter compared to dense, glob- the time at which all cells die as the death time of a species. ally connected networks which have much steeper mortality We plot the total number of species alive (out of 1000) versus curves overall (Fig. 1). For densely connected networks, we time in the middle row of Fig. 2. For the mortality rate, we also see that organisms with poor initial conditions (higher numerically compute µ(t) = −[S(t + δt) − S(t)]/[S(t)δt], η), have flatter mortality curves compared to similarly struc- where we choose the time step δt to be given as 1/4 times the tured organisms with better initial conditions (lower η). This standard deviation of the mean lifetime for each species. is because these networks are more largely affected by ini- We normalize the mortality rates by the mean mortality over tial conditions, and the probability of failure is already high time and normalize time by the mean lifetime to compare each with poor initial conditions, and does not increase much fur- case. ther with time. For very sparsely connected networks, we also The key quantity of this model is γ.A γ value close to 1 see an initial dip due to higher infant mortality rate from poor indicates that cells cooperate by secretions that enhance each initial conditions (Fig. 1a,b). As nodes are repaired, the mor- other’s survival. As we decrease γ towards 0.5 this corre- tality rate quickly decreases, and then continues to increase as sponds to cells both benefiting as well as harming each other. the organism ages. For intermediate connectivities, we see the Values of γ closer to zero describes a collection of cells that initial condition does not effect the aging curves (Fig. 1c,d). solely compete with each other, as one might expect in a For denser, globally connected networks, we do not see a large colony of non-social unicellular organisms. bioRxiv preprint doi: https://doi.org/10.1101/2021.05.27.446029; this version posted May 28, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 5

FIG. 2. Effects of hill constants and mixing factor γ on aging dynamics. We explore the mean cell population over time as well as the aging dynamics for ensembles of simulations, for varying hill constant and mixing factor parameters. For simplicity, we take kφ = kψ = k. In the top row, we plot the mean cell population over time, normalized by the mean lifetime for each system. For systems where cells communicate more through negative factors (small γ), cells die quicker at ﬁrst and then much slower later. This corresponds to shallower mortality curves, as seen in the last row. For systems where cells communicate more through cooperative factors (large γ), cells die out gradually at ﬁrst and then exhibit more of a crash. This corresponds to steeper mortality curves in the last row. As we increase the hill constant k, the effects become more pronounced. For large k, cooperative systems age more and antagonistic systems age even less. Survival curves S(t) in the middle row are given by recording the death times for 1000 simulations for each parameter set. Mortality curves were calculated using 1/4 the standard deviation of death times as a time step (δt) and taking a discrete derivative of the survival function (µ(t) ≈ −[S(t + δt) − S(t)]/[S(t)δt]) and then normalized by mean mortality rate over time. Other parameters were set to Dφ = Dψ = 5, v = 0, α = 100, λφ = λψ = 50, and Aφ = Aψ = 10.

As expected, we see that the mortality curves become The nature of the interaction also manifests in the cell pop- steeper as we increase γ. This is because as we increase γ, ulation dynamics. In the top row of Fig. 2, we plot the mean the interaction between cells go from being deleterious to co- cell population versus time, normalized by mean lifetime. We operative. When cells are strongly dependent on each other see for more cooperative systems (higher γ), cells die slower for survival, local failures can quickly spread to global ones, at first and then have more of a “crash” near the end of the leading to system wide collapse. For low γ on the other hand, organism’s lifetime. Whereas for more competitive systems cells influence each other negatively. As some cells begin to (lower γ), cells die quickly at first, and then slowly fizzle out die, there is less waste pollution and others can survive longer. towards the end of the organism’s lifetime. The mortality curves are therefore more shallow for low γ. As we increase the hill constant k, we see the same trends bioRxiv preprint doi: https://doi.org/10.1101/2021.05.27.446029; this version posted May 28, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 6

stant) is to exaggerate the aging behaviour for both systems. With higher flow, the secreted factors are transported further away from the cell producing them, allowing more cells to become coupled together. As with diffusion, we again see larger aging for cooperative systems and less for competitive ones. We therefore find that how signals and factors are transported also strongly influences the aging dynamics of an organism. With these insights, we compare the results of our model to the mortality curves of various species in the next section. FIG. 3. Coupled effects of diffusion and flow on purely cooperative and antagonistic systems. On the left we plot the mortality, normalized by the mean mortality, at the mean lifetime, versus diffusion Fits to empirical data rate, setting flow velocity to v = 0. We see that as we increase diffusion, more cells are able to couple together, thus exaggerating the aging dynamics. The cooperative system (γ = 1) ages more and the We next compare mortality curves obtained from our diffu- antagonistic system (γ = 0) ages less as we increase diffusion. On sive model to those from empirical data obtained from Jones, the right we plot the same value versus flow velocity, keeping diffu- Owen R., et al. (2014) [3] in Fig. 4. As in [3], mortality sion fixed at Dφ = Dψ = 5. Increasing the velocity gives the same curves are plotted from the age of reproductive maturity up effect as increasing diffusion. Since cells are more strongly coupled, to the terminal age where only 5% of the population is alive. the aging behavior for each case becomes more pronounced. The hill Mortality values are scaled relative to their mean values for constants were set to k = k = 2 for each case. Other parameters φ ψ each species. were set to α = 100, λφ = λψ = 50, and Aφ = Aψ = 10. Phylogenetic relatedness is seen to have some correlation with the aging dynamics of species (c.f. Fig. 1 of [3]). To with the factor γ, but the effect becomes more pronounced. compare our model to empiric data, we took a representa- For larger k, the interaction between cells increases. Hence, tive species from mammals (Orcinus orca), birds (Fulmarus large γ systems age even more (steeper mortality curves) and glacialoides and Abus melba), invertebrates (Caenorhabditis low γ systems age even less (shallower mortality curves). elegans), plants (Pinus sylvestris) and reptiles (Lacerta vivpara) and fit mortality curves from our model in Fig. 4. We We therefore see that the type of interaction and the reaction compared empiric data to our model by running simulations kinetics between cells in an organism can strongly influence varying the mixing factor γ, flow velocity v, and the hill con- the mortality of an organism. Even though all organisms are stants k and k . We then compared the resulting mortality represented by the same topological system in the model, we φ ψ curves and took the best fitting curve. see that the biophysical details of the interactions also plays a major role in aging. The bird Fulmarus glacialoides has a fairly steep mortality curve with a mortality rate at terminal age (33 years) at around 6 times the average mortality. Our best fit model for Fulmarus glacialoides is given by the values v = 150, kφ = 1, kψ = 1, Effects of diffusion and flow and γ = 0.15 (Fig. 4a). The mammal Orcinus orca has a slighly less steep mortality curve with a mortality at termi- We next study the effects of varying diffusion and flow rates nal age (59 years) at around 5 times the average. Our best on the aging dynamics of organisms. As diffusion and flow fit model for Orcinus orca is given by the values v = 150, rates will alter the extent to which cells interact and couple to kφ = 2, kψ = 2, and γ = 0.125 (Fig. 4b). The inverte- each other, we expect the aging dynamics of a species to also brate Caenorhabditis elegans has a mortality at terminal age vary with different flow and diffusion rates. (25 days) at around 3 times the average mortality. We fit our For simplicity, we focus on purely cooperative (γ = 1) and model with values v = 150, kφ = 3, kψ = 3, and γ = 0.075 purely antagonistic (γ = 0) interactions. To compare the ag- in Fig. 4c. The bird Apus melba has a much smaller mor- ing dynamics for varying diffusion and flow rates, we plot the tality at terminal age (16 years) compared to the fulmar, at mortality at the mean lifetime of a system versus different dif- around 2 times the average mortality. We fit our model with fusion rates and flow rates in Fig. 3. values v = 0, kφ = 1, kψ = 3, and γ = 0.02 in Fig. 4d. When we increase the diffusion rate, keeping flow at zero, The plant Pinus sylvestris has a mortality at terminal age (30 we see that the effect of increased diffusion is to exaggerate years) at around twice the average mortality. We fit our model the aging behavior of competitive (γ = 0) and cooperative with values v = 0, kφ = 1, kψ = 3, and γ = 0.01 in Fig. (γ = 1) systems. Larger diffusion couples more cells to- 4e. Finally, the reptile Lacerta vivpara (common lizard) has a gether, as the secreted factors (negative or positive) diffuse out roughly flat mortality curve with mortality rate at terminal age further. Hence, larger diffusion leads to a larger effect, steeper (6 years) just slighly above the average mortality. We fit our mortality curves for cooperative systems and shallower curves model with values v = 150, kφ = 1, kψ = 2, and γ = 0.0. in for competitive ones. Fig. 4e. We see the same behavior when we increase flow. We see in In general mammals tend to have the most drastic aging be- Fig. 3 that the effect of increased flow (keeping diffusion con- havior, reptiles tend to age less (some may even exhibit neg- bioRxiv preprint doi: https://doi.org/10.1101/2021.05.27.446029; this version posted May 28, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 7

FIG. 4. Mortality curves obtained from empirical data and fitted with theoretical model. Empirical mortality data was obtained from [3]. Mortality curves are plotted as functions of age from the age of reproductive maturity up to the age where 5% of the population is still alive. Mortality values are normalized by the mean value for each case. a, The bird Fulmarus glacialoides has a steep mortality curve, with mortality at terminal age at around 6 times the average level of adult mortality. Fit parameters are v = 150, kφ = 1, kψ = 1, and γ = 0.15. b, The mammal Orcinus orca have a slighly less steep mortality curve with the mortality at terminal age around 5 times the average level of adult mortality. Fit parameters were v = 150, kφ = 2, kψ = 2, and γ = 0.125. c, Caenorhabditis elegans, an invertibrate, has an even less steep mortality curve, with the mortality at terminal age at around 2 times the average mortality. Fit parameters are v = 150, kφ = 3, kψ = 3, and γ = 0.075. d, The bird Apus melba, has mortality at terminal age at around 3 times the average mortality. Fit parameters are v = 0, kφ = 1, kψ = 3, and γ = 0.02. e, Mortality curve and fit for Pinus sylvestris (Scotts pine). Fit parameters are v = 0, kφ = 1, kψ = 3, and γ = 0.01. f, Mortality curve and fit for the lizard Lacerta vivipara. Fit parameters are v = 150, kφ = 1, kψ = 2, and γ = 0.0. Other parameters were set to Dφ = Dψ = 5, λφ = λψ = 50, and Aφ = Aψ = 10. The time constant was set to α = 100 for all simulations and the mortality curves were rescaled to have the same time at terminal age.

ligible or negative senescence), and plants seem to age even bination of factors such as metabolism and thermal regula- less. Whereas, birds and invertebrates are more diverse in tion, regenerative capabilities [22], fertility and life cycles their aging behavior [3]. Here we fit our model varying pa- [23–25], and modular structure [26], due to both evolution- rameters for the mixing factor γ, and hill constants for each ary and mechanical differences across species. There have type of factor kφ and kψ. We see that we are able to find good been a few theoretical studies aiming to explain the diversity fits to the empiric data by varying these parameters, which of aging. Indeterminate growth has been proposed as a factor correspond to how strongly cells may depend on each other leading to constant or negative senescence. If mortality de- within the organism. creases with increasing size and reproductive potential rises, We see that species that age less, with shallower mortal- non-senescence can become optimal evolutionarily [2]. Inde- ity curves, are fit with smaller values for the γ factor, cor- terminate growth may then explain why plants and some rep- responding to less cooperative interactions between compo- tiles seem to age slower than species with determinate growth. nents. Therefore, we see that the biophysical details of the Here we consider an additional, possibly alternative mecha- interactions between cells may contribute to the diversity in nism behind the phylogenetic trends in aging: We hypothesize aging across different species. that how various factors are transported between cells determines the nature of µ(t). We studied two separate models to explore the effects of in- DISCUSSION terdependencies on aging. In the first model, we emphasized the connectivity structure, and study aging processes on en- It has been proposed that flatter mortality curves (such as sembles of different network topologies. We found in general that of reptiles and amphibians) [3] may be due to a com- that sparser, locally connected topologies give rise to shal- bioRxiv preprint doi: https://doi.org/10.1101/2021.05.27.446029; this version posted May 28, 2021. The copyright holder for this preprint (which was not certified by peer review) is the author/funder. All rights reserved. No reuse allowed without permission. 8

lower mortality curves, whereas denser, globally connected compete more for small γ, leading to more steep or shallow networks give steeper mortality curves. We also found the mortality curves, respectively. initial conditions more largely affect denser networks. Poor Finally, we compared the results of our theoretical model initial conditions greatly increase the probability of failure by fitting mortality curves to empirical data obtained from for denser networks, with little room to increase over time. Jones, Owen R., et al. (2014) [3]. We plotted empirical mor- This then leads to shallower mortality curves (Fig. 1e-h). For tality curves and theoretical fits for mammals (Orcinus orca), sparser networks, we find a larger repair rate can lead to shal- birds (Fulmarus glacialoides and Apus melba), invertebrates lower mortality curves (Fig. 1c,d), as opposed to denser net- (Caenorhabditis elegans), plants (Pinus sylvestris), and rep- works where the repair rate has little effect. As failure prop- tiles (Lacerta vivpara) in Fig. 4. Fit values for γ corre- agates slower in sparser networks, they are more amenable to sponded to larger γ for species with steeper mortality curves repair. and smaller γ for shallower ones, as expected. We there- In the second model, we emphasized the biophysical prop- fore find that the biophysical details of how various factors erties of the signals being communicated. In this diffusion- are transported within an organism can strongly determine its advection model, cells influence each other’s function through aging dynamics. secreted factors that are able to diffuse and flow. In contrast In reality, there may be many diffusive factors that deter- to network models, we studied a simple circular topology, but mine the aging dynamics of an organism. These may also be were able to focus on the inter-cellular biophysical processes integrated in various ways beyond what we consider here. We in detail. Even with this simple topology, we find that differ- expect that multiple factors with similar diffusion-advection ences in the types of factors and how those factors propagate length scales would not significantly change our results, and can strongly determine the mortality curves of a species. the factors φ and ψ considered here can be taken as a sum of We assumed cells communicate through two types of se- such factors or as limiting factors. Exploring different ways creted factors, a cooperative factor φ such as antioxidant en- of integrating factors may give new and interesting results be- zymes or growth or survival factors [18–21], and an antago- yond what we consider here. nistic factor ψ such as waste and apoptotic factors [27, 28]. We also took in our model, for simplicity, each type of fac- We modeled the relative ratio of factors through the mixing tor to be produced by all cells at a constant rate. In reality, parameter γ, with γ = 1 corresponding to a purely coopera- the production of factors may depend on the physiolgoical tive system and γ = 0 a purely antagonistic one. state of the cell or on some feedback mechanism. Factors pro- We found the aging behavior of the system is largely de- duced by cell death or lysis might also be relevant to include. termined by the factor γ as well as by the hill constants kφ Local, non-diffusive compounds such as lipofuscin [29] may and kψ. Lower values of γ, corresponding to less cooperative also contribute to aging [30]. The inclusion of these additional interactions between cells in a species, give shallower mor- types of factors might lead to interesting future research. tality curves, and higher values of γ give steeper mortality For further investigation, we take can also take into account curves (Fig. 2). This is because for larger γ, cells depend on cellular reproduction and repair in the diffusion-advection each other for survival. As damages accumulate, a late life model. This may produce flatter and even negatively aging “crash” becomes more likely, where all cells die out, killing mortality curves as seen in some plants and reptiles. It would the organism. For smaller γ, cells cooperate less and may also be interesting to see the effect cell health has on flow. For even compete with each other. Accumulated damages then do example, the integrity and tension of the extra-cellular matrix not affect other cells as much and system wide death is less which helps to regulate the interstitial fluid pressure depends drastic. Larger values for the hill constants further exagger- on cellular viability [31]. It may be insightful to also include ate these effects, giving a larger difference between small and the mechanical interplay of flow and cell viability in a more large values of γ when the hill constants are also larger (Fig. complete theory of aging. 2). Finally, it could be interesting to combine the two models We also explored the effects of diffusion and flow on aging. explored here, into a biophysically realistic network approach, We found larger diffusion and flow also exaggerate aging ef- extending our current one-dimensional diffusion-advection fects (Fig. 3). A larger diffusion and/or flow rate couples more model to more realistic topologies. This approach may also distant cells together. Cells then cooperate more for large γ or give further insight into the aging dynamics of various species.

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